Twisted regular representations of vertex operator algebras

This paper is to study what we call twisted regular representations for vertex operator algebras. Let $V$ be a vertex operator algebra, let $σ_1,σ_2$ be commuting finite-order automorphisms of $V$ and let $σ=(σ_1σ_2)^{-1}$. Among the main results, for any $σ$-twisted $V$-module $W$ and any nonzero complex number $z$, we construct a weak $σ_1\otimes σ_2$-twisted $V\otimes V$-module $\mathfrak{D}_{σ_1,σ_2}^{(z)}(W)$ inside $W^{*}$. Let $W_1,W_2$ be $σ_1$-twisted, $σ_2$-twisted $V$-modules, respectively. We show that $P(z)$-intertwining maps from $W_1\otimes W_2$ to $W^{*}$ are the same as homomorphisms of weak $σ_1\otimes σ_2$-twisted $V\otimes V$-modules from $W_1\otimes W_2$ into $\mathfrak{D}_{σ_1,σ_2}^{(z)}(W)$. We also show that a $P(z)$-intertwining map from $W_1\otimes W_2$ to $W^{*}$ is equivalent to an intertwining operator of type $\binom{W'}{W_1\; W_2}$, which is a twisted version of a result of Huang and Lepowsky. Finally, we show that for each $τ$-twisted $V$-module $M$ with $τ$ any finite-order automorphism of $V$, the coefficients of the $q$-graded trace function lie in $\mathfrak{D}_{τ,τ^{-1}}^{(-1)}(V)$, which generate a $τ\otimes τ^{-1}$-twisted $V\otimes V$-submodule isomorphic to $M\otimes M'$.